Linear system

A general deterministic system can be described by an operator, H, that maps an input, x(t), as a function of t to an output, y(t), a type of black box description. Linear systems satisfy the property of superposition. Given two valid inputs

${\displaystyle x_{1}(t)}$

${\displaystyle x_{2}(t)}$

as well as their respective outputs

${\displaystyle y_{1}(t)=H\left\{x_{1}(t)\right\}}$

${\displaystyle y_{2}(t)=H\left\{x_{2}(t)\right\}}$

then a linear system must satisfy

${\displaystyle \alpha y_{1}(t)+\beta y_{2}(t)=H\left\{\alpha x_{1}(t)+\beta x_{2}(t)\right\}}$

for any scalar values α and β.

The system is then defined by the equation H(x(t)) = y(t), where y(t) is some arbitrary function of time, and x(t) is the system state. Given y(t) and H, the system can be solved for x(t). For example, a simple harmonic oscillator obeys the differential equation:

${\displaystyle m{\frac {d^{2}(x)}{dt^{2}}}=-kx}$.

If

${\displaystyle H(x(t))=m{\frac {d^{2}(x(t))}{dt^{2}}}+kx(t)}$,

then H is a linear operator. Letting y(t) = 0, we can rewrite the differential equation as H(x(t)) = y(t), which shows that a simple harmonic oscillator is a linear system.

The behavior of the resulting system subjected to a complex input can be described as a sum of responses to simpler inputs. In nonlinear systems, there is no such relation. This mathematical property makes the solution of modelling equations simpler than many nonlinear systems. For time-invariant systems this is the basis of the impulse response or the frequency response methods (see LTI system theory), which describe a general input function x(t) in terms of unit impulses or frequency components.

Typical differential equations of linear time-invariant systems are well adapted to analysis using the Laplace transform in the continuous case, and the Z-transform in the discrete case (especially in computer implementations).

Another perspective is that solutions to linear systems comprise a system of functions which act like vectors in the geometric sense.

A common use of linear models is to describe a nonlinear system by linearization. This is usually done for mathematical convenience.

The time-varying impulse response h(t₂, t₁) of a linear system is defined as the response of the system at time t = t₂ to a single impulse applied at time t = t₁. In other words, if the input x(t) to a linear system is

${\displaystyle x(t)=\delta (t-t_{1})}$

where δ(t) represents the Dirac delta function, and the corresponding response y(t) of the system is

${\displaystyle y(t)|_{t=t_{2}}=h(t_{2},t_{1})}$

then the function h(t₂, t₁) is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the following causality condition must be satisfied:

${\displaystyle h(t_{2},t_{1})=0,t_{2}<t_{1}}$

The output of any general continuous-time linear system is related to the input by an integral which may be written over a doubly infinite range because of the causality condition:

${\displaystyle y(t)=\int _{-\infty }^{t}h(t,t')x(t')dt'=\int _{-\infty }^{\infty }h(t,t')x(t')dt'}$

If the properties of the system do not depend on the time at which it is operated then it is said to be time-invariant and h is a function only of the time difference τ = t − t' which is zero for τ < 0 (namely t < t' ). By redefinition of h it is then possible to write the input-output relation equivalently in any of the ways,

${\displaystyle y(t)=\int _{-\infty }^{t}h(t-t')x(t')dt'=\int _{-\infty }^{\infty }h(t-t')x(t')dt'=\int _{-\infty }^{\infty }h(\tau )x(t-\tau )d\tau =\int _{0}^{\infty }h(\tau )x(t-\tau )d\tau }$

Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called the transfer function which is:

${\displaystyle H(s)=\int _{0}^{\infty }h(t)e^{-st}\,dt.}$

In applications this is usually a rational algebraic function of s. Because h(t) is zero for negative t, the integral may equally be written over the doubly infinite range and putting s = iω follows the formula for the frequency response function:

${\displaystyle H(i\omega )=\int _{-\infty }^{\infty }h(t)e^{-i\omega t}dt}$

The output of any discrete time linear system is related to the input by the time-varying convolution sum:

${\displaystyle y[n]=\sum _{m=-\infty }^{n}{h[n,m]x[m]}=\sum _{m=-\infty }^{\infty }{h[n,m]x[m]}}$

or equivalently for a time-invariant system on redefining h(),

${\displaystyle y[n]=\sum _{k=0}^{\infty }{h[k]x[n-k]}=\sum _{k=-\infty }^{\infty }{h[k]x[n-k]}}$

where

${\displaystyle k=n-m\,}$

represents the lag time between the stimulus at time m and the response at time n.

• Linear system of divisors in algebraic geometry
• Shift invariant system
• Linear time-invariant system
• Nonlinear system
• System analysis
• System of linear equations